Indiana Law Journal Indiana Law Journal

Volume 55 Issue 3 Article 3

Spring 1980

On the Use of Statistics in Employment Discrimination Cases On the Use of Statistics in Employment Discrimination Cases

Richard M. Cohn University of Wisconsin- Madison

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Recommended Citation Recommended Citation Cohn, Richard M. (1980) "On the Use of Statistics in Employment Discrimination Cases," Indiana Law Journal: Vol. 55 : Iss. 3 , Article 3. Available at: https://www.repository.law.indiana.edu/ilj/vol55/iss3/3

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On the Use of Statistics in EmploymentDiscrimination Cases

RICHARD M. COHN*

The use of quantitative information in employment discrimina-tion cases has become increasingly common.' Differences in ratesof hiring, promotion, demotion or dismissal between minority andmajority groups are used to develop a prima facie case of discrimi-nation. Evidence of this type is deemed to represent the outcomeof employers' discriminatory practices, including practices thedescription of which does not indicate any overt discriminatory in-tent.2 This reliance on quantitative information undoubtedly willincrease since the new Uniform Guidelines on Employee SelectionProcedures (1978) (U.G.E.S.P.) specifically define evidence of dis-crimination in quantitative terms.'

As a result of this increased emphasis on quantitative informa-tion, it is necessary that litigants in discrimination cases, includinggovernmental agencies, become intelligent consumers of such'infor-mation. Contrary to popular opinion, numbers rarely speak forthemselves. Both interpretation and evaluation are required of allbut the most elementary quantitative descriptions and analyses. Asevidence of this, this article discusses three issues concerning thepotential misuse of quantitative information in employment dis-crimination cases. Each issue illustrates a basic property of quanti-tative information which affects its probative value. First, the basicissue of when statistical inference is appropriate is considered. To

* A.B. 1970, M.A. 1975, Ph. D. 1977, University of Michigan. Member of the Department

of Sociology, and the Center for Demography and Ecology, University of Wisconsin-Madison.

' For reviews of the use of quantitative evidence in employment discrimination cases, seeHolley & Feild, Using Statistics in Employment Discrimination Cases, 4 EMPLOYEE REL.L.J. 43 (1978); Joseph, Evidence: Statistical Proof in Employment Discrimination Cases, 28OKLA. L. REV. 885 (1975); Montlack, Using Statistical Evidence to Enforce the LawsAgainst Discrimination, 22 CLEv. ST. L. REv. 259 (1973).

1 As an example of the acceptance of quantitative evidence of discrimination where thedescription of an employer's hiring policy does not indicate any overt discriminatory prac-tice, see Lea v. Cone Mills Corp., 301 F. Supp. 97 (M.D.N.C. 1969).

1 43 Fed. Reg. 38,295 (1978) [hereinafter cited as U.G.E.S.P.]. Section 3A of theU.G.E.S.P. states: "Procedure having adverse impact constitutes discrimination unless justi-fied." Id. at 38,297. Section 4D defines adverse impact as "[a] selection rate for any race,sex, or ethnic group which is less than four-fifths (4/5) (or eighty percent) of the rate for thegroup with the highest rate will generally be regarded by the Federal enforcement agenciesas evidence of adverse impact." Id. Section 16R defines selection rate as "[t]he proportion ofapplicants or candidates who are hired, promoted, or otherwise selected." Id. at 38,308.

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illustrate the confusion concerning this issue, a recently publishedapplication of an elementary statistical test to employment selec-tion procedures is critically reviewed. The second issue consideredis the inappropriate use of particular quantitative measures as evi-dence of discrimination. Illustrative of this is the new U.G.E.S.P.strategy for determining "prosecutorial discretion" in employmentdiscrimination cases. A simple quantitative model of an employer'shiring rate is developed to indicate the problematic aspects of theU.G.E.S.P. strategy. The final issue discussed is how quantitativeanalysis may be used to distinguish when a characteristic, such assex or age, is a "bona fide occupational qualification" from when itis the basis of discrimination.

THE INAPPROPRIATE USE OF STATISTICS AND STATISTICAL

INFERENCE

An important misuse of quantitative information in employmentdiscrimination cases may result from the failure to distinguishwhen such information is a "statistic" as opposed to a "parame-ter." A statistic is quantitative information which is calculatedfrom sample data. It is used to estimate the corresponding valuefor the entire population. This value for the population is the pa-rameter. The estimation of the parameter from the statistic is aninferential process. Statistical inference is a probability statement;a statement that a known value of a sample statistic equals the(unmeasured) population parameter with a certain probability. Forexample, if the results of a sample survey are statistics which indi-cate women earn less than men, it can be inferred, with a givenlevel of certainty, that in the entire population women, on average,earn less than men. That women actually do earn less than men inthe entire population can never be known with complete certainty,i.e., the probability of the truth of the statement is less than 1.0,unless a complete census is carried out.

For a statistic to be an accurate estimate of a parameter, thesample on which the statistic is based must have certain proper-ties. The most important properties are that individuals in thepopulation have a fixed, known and nonzero probability of beingincluded in the sample, and that the sample is representative ofthe population.4 When these properties are absent, the use of a

Typically these properties of a sample are insured by some form of random selectionprocess. Although sampling procedures can be considerably more complicated, the classic

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statistic to infer the value of the parameter is inappropriate.When using any quantitative information, two basic characteris-

tics of the information must be known. First, it is necessary toknow if the information is truly statistical, i.e., calculated from asample, or if the information is calculated from a complete count(census) of some population. For example, information on maleand female promotion rates in a company are parameter values forthat company and cannot be viewed as statistics to infer differ-ences in promotion rates for all companies. The specific company'spromotion rate information does not constitute a statistic for asample of companies because the probability of any other com-pany's promotion rates being included as data is zero. Given theinformation about one company's promotion rates, it is not possi-ble to infer differences in the industry or any other aggregate. Thecompany's promotion rate is a parameter value for that company,known with complete certainty. The information is not a statisticwhich can be used to estimate the parameter value for any otheraggregate with any known certainty.

The second property of any quantitative information which mustbe known is the definition of the population to which the informa-tion applies. In the above example, the population is defined as theemployees of the specific company. In situations where the infor-mation is truly statistical, the population to which statistical infer-ences can be made must be precisely defined. For example, if thesample was selected to include only labor force participants, thenthe population to which inferences can be made is the populationof labor force participants, not the total population in a certaingeographic area. It is inappropriate to use statistics calculatedfrom this sample to infer parameter values to any populationlarger or smaller than the population of labor force participants.

An example of the confusion which can result from the failure todetermine whether quantitative information is a statistic or a pa-rameter is a recent Harvard Law Review article which describes anelementary statistical test purported to be useful in indicating theadverse impact of employment testing.5 In the article, the author,

example of drawing names from a hat to produce a sample is instructive. Having each indi-vidual's name appear on only one slip insures that each member of the population will havea fixed, known, and nonzero probability of being selected (in this case each member has anequal chance of selection). Thoroughly stirring the names and selecting them blindly insuresthat the sample will be representative of the population. Details on the necessary propertiesof a sample and descriptions of sampling techniques can be found in most elementary statis-tics texts and research methods texts. See, e.g., L. KIsH, SURVEY SAMPLING (1965).

' Shoben, Differential Pass-Fail Rates in Employment Testing: Statistical Proof Under

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Professor Elaine Shoben, proposes that the difference in passingrates on an employment selection test between minority and ma-jority applicants be considered a statistic and used to estimatewhether the test has an adverse impact in the population.6 Thisproposed use of the difference in passing rates information for aspecific employer as a statistic is contrasted with the U.G.E.S.P.'s"4/5ths rule" to indicate adverse impact of the employment test.With the guideline rule, a difference of twenty percent or more inthe proportion passing the test between minority and majority ap-plicants is viewed as indicating adverse impact.

The Shoben article argues that the proposed statistical test issuperior to the use of the "4/5ths rule" which is not liable to statis-tical inference. While the U.G.E.S.P. indicate that the "4/5thsrule" is to be used in lieu of statistical tests, the guidelines arevague as to when, if ever, the "4/5ths rule" is to be applied. Asnoted in section 4D of the U.G.E.S.P., a statistically significant dif-ference in passing rates smaller than 0.2 may be used as evidenceof adverse impact when there is a large number of applicants. Incases where there is a small number of applicants, a difference inpassing rates greater than 0.2 may not be used to indicate adverseimpact if the difference is not statistically significant.' Thus, thenumber of applicants an employer processes may determinewhether or not adverse impact of the selection procedures is indi-cated, regardless of the true discriminatory nature of the test, be-

Title VII, 91 HARV. L. REv. 793 (1978). Adverse impact, as used in the Shoben article anddefined in § 16B of the U.G.E.S.P., is "[a] substantially different rate of selection ii, hiring,promotion, or other employment decision which works to the disadvantage of members of arace, sex, or ethnic group." 43 Fed. Reg. 38,307 (1978).

6 The Shoben article suggests the use of the difference in proportions test, where theobserved difference in proportions passing the test between minority and majority appli-cants of an employer is used to estimate whether for the entire population there exists adifference in passing rates. For a description of the test, see Shoben, supra note 5, at 797-805, or any elementary statistics text.

7 See note 3 supra.8 Section 4D of the U.G.E.S.P. states:

Smaller differences [than 0.2] in selection rate may nevertheless constitute ad-verse impact, where they are significant in both statistical and practical termsor where the user's actions have discouraged applicants disproportionately ongrounds of race, sex, or ethnic group. Greater differences in selection rate maynot constitute adverse impact where the differences are based on small num-bers and are not statistically significant . ...

43 Fed. Reg. 38,297 (1978). See also Questions and Answers 22 & 24, interpreting and clari-fying the U.G.E.S.P. 44 Fed. Reg. 11,999 (1979). As the federal enforcement agency hasdiscretion in choosing whether a test of statistical significance or the "4/5ths rule" is used toindicate adverse impact, considerable uncertainty among employers as to when they are inviolation of the U.G.E.S.P. can be expected.

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cause alternate decision rules may be applied to the quantitativeinformation.

Any statistical test, the one proposed in the Shoben article orthe test proposed as an alternative to the "4/5ths rule" in theU.G.E.S.P., is inappropriate to indicate adverse impact. The em-ployer's quantitative information on passing rates is not a statistic,but rather a population parameter. As such, the use of any test ofstatistical significance of the differences in passing rates to infer toa population larger than that of the individuals who took the testis inappropriate. In the Shoben article it is argued that the test-takers can be considered a sample of all potential job applicants inthe employer's "labor pool," i.e., they are a sample of a populationdefined as the potential job applicants of the employer. In fact, theapplicants who took the test are not a sample of anything, forthere is no fixed or known probability by which all potential appli-cants become test-takers, i.e., become part of the sample. Thoseindividuals who took the test define the population of test-takers.No statistical inference relating a difference in their observed pass-ing rates to an unmeasured difference in passing rates in a largerpopulation is warranted.'

A similar argument applies to the U.G.E.S.P.'s suggested use ofstatistical tests as an alternative to the "4/5ths rule" in determin-ing adverse impact. The information of the employer's passingrates is not sample information, it is a population parameter.1 °

There is no appropriate use of a test of statistical significance withthis information because there is no larger population to which toinfer. Regardless of the number of applicants taking the test, thedifference in passing rates between minority and majority appli-cants has the same value in indicating adverse impact.11 As an al-

9 Shoben, supra note 5, at 801, notes that as a "sample," the actual test-takers may notbe representative of the "population" of all potential test-takers. This is likely true, but it isirrelevant given that the test-takers do not constitute a sample in the first place.1o The definition of the population need not be limited to only those applicants taking the

test at a single administration. The population may be defined as all the applicants takingthe test during a specific year. For purposes of evaluating adverse impact of the test, achange in the contents of the test would require a new population of applicants be defined.11 With statistical information, the certainty with which one can infer that the sample

statistic equals the population parameter increases with the size of the sample. Thus, whenquantitative information is truly statistical, its acceptance by the court or governmentalenforcement agency may be questioned if the sample size is deemed inadequate. Quantita-tive information which is calculated from an entire population cannot be disputed on thebasis of the number of observations. Thus, in the above example of the difference in passingrates to indicate adverse impact, because the information is a population parameter, thenumber of observations (test-takers) is irrelevant t6 the question of adverse impact. The

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ternative to the "4/5ths rule," the use of a test of the statisticalsignificance of the difference in passing rates is inappropriate. 2

The level of statistical significance indicates the certainty withwhich a difference in passing rates observed in a sample can beinferred to exist in a population.'8 Statistical significance is mean-ingless when not referring to the relation of a sample value (statis-tic) to a population value (parameter). The failure to note that theinformation obtained from an employer is not sample informationcan produce erroneous uses of quantitative information such asfound in the Shoben article and section 4D of the U.G.E.S.P.

The determination of whether the information is derived from asample is relevant to the quality of the evidence indicating dis-crimination. If the information on the test-takers' performance iserroneously conceptualized as being sample data from a populationof all potential test-takers, the test of statistical significance ad-dresses the question: "Is it likely there would be such a differencein observed test scores if the test has no adverse impact in theentire population?" Not only are the observed scores inadequateevidence to answer this question when they are viewed as sampleresults, due to the unknown bias of the "sample," but this questionis not the legal question at issue. The question at issue is whether

census of Liechtenstein is as informative as the census of China.However, there is a reason for requiring a minimum number of test-takers in the popula-

tion which is unrelated to the issue of statistical significance. Since several factors, otherthan race, may determine test performance, in comparing the passing rates of minority andmajority applicants, it is important to control for racial differences in these nonracial fac-tors, e.g., education, experience. If there are racial differences in these factors, there shouldbe an adequate number of observations of minority and majority group applicants with sim-ilar qualifications to allow for racial comparisons when these qualifications are held con-stant. The importance of controlling for racial differences in job qualifications when evaluat-ing racial differences in hiring is discussed later in this article. See notes 25-38 &accompanying text infra. A

12 Rather than not applying the "4/5ths rule" because it is too "stringent" for employerswith few applicants and too "lenient" for employers with many applicants, an alternativeform of the rule might be used. If federal enforcement agencies consider it appropriate toinclude the number of applicants, as well as the difference in selection rates, in determiningadverse impact, a graduated "rule" of the difference in selection rates is appropriate. Thegraduated structure of the rule could be progressive, similar to the structure of the incometax rate. Thus, the larger the number of applicants processed by an employer, the smallerthe difference in selection rates necessary to indicate adverse impact.

13 Actually, the common manner by which the level of statistical significance is stateddenotes the degree of certainty of no difference not existing in the population. Thus, the0.05 level of significance indicates a certainty of 1 out of 20 that no difference in selectionrates does not exist in the population. The reason for the double negative is that statisticaltests are formulated as a rejection of a (null) hypothesis which is counter to the actualhypothesis of interest.

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or not the company's test has produced a differential passing ratebetween groups. This question is nonprobabilistic, has no referenceto test-takers at any other point in time, and is unequivocally an-swered by treating the quantitative information as populationdata.

DEFINING DISCRIMINATION WITH AN INAPPROPRIATE MEASURE

The second misuse of quantitative information in employmentdiscrimination cases is the inappropriate choice of a particularmeasure as evidence of discrimination. Various ratios have beenused to indicate disproportionate hiring or employment of minor-ity group members. Perhaps the most commonly used parameter isthe ratio of the proportion of hires (H) in the population or laborpool (L) between minority and majority groups, i.e., [(H/L)blacks/(H/L)whites]. A value of this ratio of less than one has been usedas evidence of discrimination.1'

The use of this ratio as evidence of an employer's discriminatorypractices can be misleading. The proportion hired for a particulargroup can be decomposed as the product of several proportions:

(H/L) = (HIS) (SIQ) (Q/A) (AlL)

where:L-number of group members in the labor poolA=number of applicantsQ=number of applicants who are qualified for the jobS=number of qualified applicants who pass the employer's se-

lectibn proceduresH-number who are hired.

Ratios of each of these proportions for minority and majoritygroups can be indicated as[(H/L)blacks/(H/L)whites] - [(H/S)blacks/(H/S)whites] [(SIQ)blacks/(S/Q)whites] [(Q/A)blacks/(Q/A)whites] [(A/L)blacks/(A/

14 See, e.g., Boston Chapter, NAACP v. Beecher, 504 F.2d 1017 (1st Cir. 1974); Ochoa v.Monsanto Co., 473 F.2d 318 (5th Cir. 1973); Castro v. Beecher, 459 F.2d 725 (1st Cir. 1972);Parham v. Southwestern Bell Tel. Co., 433 F.2d 421 (8th Cir. 1970). In the above cases, thetotal population, rather than the size of the relevant labor pool, was used to indicate L. Theinappropriateness of using the total population instead of a measure of the size of the laborpool is noted in Hazelwood School- Dist. v. United States, 433 U.S. 299, 308 n.13, 310-12(1977).

To be consistent with the form of the measure used in court cases and the U.G.E.S.P., Hrepresents the number of individuals who are hired by the company. A more precise indica-tor of an employer's discriminatory practices would be obtained if H stood for the numberof individuals who are offered employment, regardless of whether they accept the offer.

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L)whites].The ratio of the proportions (A/L) indicates the difference betweenminority and majority groups in their application for the job. Theratio of the (A/L) proportions can be assumed to be only partiallyinfluenced by the employer's recruitment efforts. A series of otherfactors, including alternative job opportunities in the labor market,may differentially determine the proportion (A/L), and conse-quently (H/L), for minority and majority groups. Thus the use ofthe ratio of the proportions (H/L) to indicate the employer's dis-criminatory practices, without also indicating the proportions (A/L), can be misleading.15

Perhaps recognizing the fallibility of this measure, theU.G.E.S.P. indicate that the appropriate indicator of discrimina-tory practice is not the ratio of the hiring proportions (H/L), butthe ratio of the "selection rates."' 6 Using the notation describedabove, the selection rate may be indicated as (H/A) and decom-posed as (H/A) = (H/S)(S/Q)(Q/A).1 7

The rate at which minority applicants are selected, i.e., (H/A)blacks, is compared to the rate for majority group applicants,(H/A)whites. If the ratio of these rates is less than 0.8 the "4/5thsrule" is violated and adverse impact of the employer's selectionprocedures is indicated. While the selection rate (H/A) is a moreaccurate measure of the employer's discriminatory practices thanthe hiring rate (H/L), it too can be misleading. To show this, thedeterminants of the selection rate's components must beconsidered.

The proportion of applicants truly qualified for the job, (Q/A), isgenerally unknown to the employer.' 8 The employer has little or nocontrol over this proportion. There is no reason to assume that theproportion (Q/A) is the same for minority and majority groups.Due to unequal opportunities for schooling, job training programs,accumulation of relevant job experience, and other factors, it ispossible that the minority group has a lower (Q/A) proportion than

5 The misleading nature of using the ratio of proportions (H/L) without also reporting

the ratio of the proportions (A/L) has been noted by the courts. See Richardson v. IndianaBell Tel. Co., 2 Fair Empl. Prac. Cas. (BNA) 797 (S.D. Ind. 1970).

le See note 3 supra, for the U.G.E.S.P. definition of "selection rate."17 For definitions of what constitutes an "applicant" and a "candidate," see Question and

Answer 15, interpreting the U.G.E.S.P. 44 Fed. Reg. 11,998 (1979)."8 Unless an analysis of the job produces a definitive list of the qualifications for the

particular job and these qualifications are characteristics of applicants which are vested inthe applicants, rather than acquired after employment, the rate (Q/A) can never be knownwith certainty.

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the majority group. Conversely, it is equally possible that becauseof other employers refusing to hire qualified minority workers,there is a surplus of qualified minority applicants for a specific em-ployer and therefore the (Q/A) proportion of the minority group isgreater than the (Q/A) proportion of the majority group.

The proportion of qualified applicants who pass the employer's,selection procedures, (S/Q), can be assumed to be equal for minor-ity and majority groups if the procedures are fairly administeredand actually measure the qualifications for the job. 19 Since thenumber of qualified applicants, Q, is unknown, the ratio of the (SIQ) proportions of the groups can never be calculated directly. Yetit is a reasonable assumption, given fair and relevant selection pro-cedures, that minority and majority applicants of equal qualifica-tions will pass the selection procedures at the same rate.

The proportion of those successfully completing the selectionprocedures who are hired, (H/S), is completely determined by theemployer. This proportion is a known, directly observable quanti-ty. Any difference in the proportion between minority and majoritygroups, i.e., a ratio of proportions not equal to one, can be viewedas evidence of discrimination.20

In evaluating the adverse impact of an employer's selection pro-cedures and determining "prosecutorial discretion," the U.G.E.S.P.indicate a "bottom line" strategy will be used.21 This.strategy con-

,9 A perfectly accurate selection procedure would completely distinguish qualified fromunqualified applicants and thus (S/Q) would be equal to 1.0. The requirements for valida-tion of selection procedures described in the U.G.E.S.P. do not require such a stringent levelof validation. As indicated in § 5B of the U.G.E.S.P., "[e]vidence of the validity of a test orother selection procedure by a criterion-related validity study should consist of empiricaldata demonstrating that the selection procedure is predictive of or significantly correlatedwith important elements of job performance." 43 Fed. Reg. 38,298 (1978). Thus, the propor-tion (S/Q) may be considerably lower than 1.0. Part of the requirement for selection proce-dure validation includes producing evidence of "fairness," i.e., the selection procedure doesnot differentially indicate qualification for the job between minority and majority appli-cants. See § 14B8 U.G.E.S.P., 43 Fed. Reg. 38,299 (1978). This part of the validation processis to insure that (S/Q)blacks equals (S/Q)whites.

20 A ratio of the proportions (H/S) greater than 1.0, i.e., [(H/S)blacks/(H/S)whites] >1.0,indicates "affirmative action" on the part of the employer.

22 Section 4C of the U.G.E.S.P. states, in part,The 'bottom line.' If the information called for by sections 4A and B aboveshows that the total selection process for a job has an adverse impact, the indi-vidual components of the selection process should be evaluated for adverse im-pact. If this information shows that the total selection process does not have anadverse impact, the Federal enforcement agencies, in the exercise of their ad-ministrative and prosecutorial discretion, in usual circumstances, will not ex-pect a user to evaluate the individual components for adverse impact, or tovalidate such individual components, and will not take enforcement action

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sists of examining the ratio of the selection rates for minority andmajority groups [(H/A)blacks/(H/A)whites]. If this ratio is greaterthan 0.8, i.e., if the "4/5ths rule" is not violated, the U.G.E.S.P.indicate that the individual components of the selection procedurewill not be evaluated for adverse impact and enforcement action isnot likely. 22

The use of this "bottom line" strategy can lead either to thefalse charge of adverse impact or to the conclusion that no adverseimpact exists when, in fact, the employer's selection procedure isdiscriminatory. These results are possible because of the misuse ofthe quantitative information of selection rates (H/A) to indicateadverse impact. This problematic feature of the "bottom line"strategy may be illustrated by two examples. The cases of two hy-pothetical employers are described in Table I.

based upon adverse impact of any component of that process, including theseparate parts of a multipart selection procedure or any separate procedurethat is used as an alternative method of selection.

43 Fed. Reg. 38,297 (1978).22 Exceptions to the use of the "bottom line" strategy are noted in § 4C of the U.G.E.S.P.

The relevant part of the section states:However, in the following circumstances the Federal enforcement agencies willexpect a user to evaluate the individual components for adverse impact andmay, where appropriate, take enforcement action with respect to the individualcomponents: (1) where the selection procedure is a significant factor in thecontinuation of patterns of assignments of incumbent employees caused byprior discriminatory employment practices, (2) where the weight of court deci-sions or administrative interpretations hold that a specific procedure (such asheight or weight requirements or no-arrest records) is not job related in thesame or similar circumstances. In unusual circumstances, other than thoselisted in (1) and (2) above, the Federal enforcement agencies may request auser to evaluate the individual components for adverse impact and may, whereappropriate, take enforcement action with respect to the individualcomponent.

Id. Neither of these exceptions is relevant to the critique of the "bottom line" strategy de-scribed herein.

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TABLE I

EXAMPLES OF ERRONEOUS EVALUATIONS OF ADVERSE IMPACTUSING THE U.G.E.S.P. "BOTTOM LINE" STRATEGY

Case 1: Case 2:

False Charge of Adverse Impact Failure to Discover Adverse Impact

Whites Blacks Ratio Whites Blacks Ratio(Blacks/Whites) (Blacks/Whites)

A 1000 200 1000 200

Q 900 120 600 180

S 630 84 600 180

H 567 84 600 108

Q]A 0.90 0.60 0.67 0.60 0.90 1.50

S/Q 0.70 0.70 1.00 1.00 1.00 1.00

H/S 0.90 1.00 1.11 1.00 0.60 0.60

H/A 0.57 0.42 0.74 0.60 0.54 0.90

H/Q 0.63 0.70 1.11 1.00 0.60 0.60

where:

A = number of applicantsQ = number of applicants qualified for the jobS = number of applicants who successfully pass employer's selection proceduresH = number of applicants who are hired

H/A = "selection rate"

In Case 1, the employer would be falsely charged with an ad-verse impact of his selection procedures because the ratio of theselection rates (reported in the third column of the Table) is lessthan 0.8. Yet qualified blacks are passing the employer's selectionprocedures at the same rate as qualified white applicants and theemployer is hiring selected black applicants at a rate greater thanselected white applicants, i.e., the employer has a program of af-firmative action. The false charge of adverse impact of the selec-tion procedures results from the proportion of black applicantswho are qualified for the job being only two-thirds that of the pro-portion of white applicants.

In litigation of the charge of adverse impact, the employer inCase 1 could point to the following facts to support his case:

1. the affirmative action evident by the ratio [(H/S)blacks/(H/S)whites] equaling 1.11;

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2. qualitative evidence of fair administration of the selectionprocedures to infer (S/Q)blacks equals (S/Q)whites.

Whether the qualitative evidence of the second fact can be success-ful in countering the quantitative evidence that [(S/A)blacks/(S/A)whites] equals 0.67 (a violation of the "4/5ths rule") is not dis-cussed in the U.G.E.S.P.

In Case 2, the employer is not charged with adverse impactunder the "bottom line" strategy because the ratio of selectionrates is greater than 0.8. Yet the employer is hiring blacks whohave successfully passed the employment selection procedures at arate only sixty percent, that of whites. The failure of the "bottomline" strategy to discover this discriminatory practice is a result ofthe proportion of black applicants who are qualified for the jobbeing fifty percent greater than the proportion of whiteapplicants.23

The basic flaw of the U.G.E.S.P.'s "bottom line" strategy is thatit uses too highly aggregated quantitative information to indicatediscrimination. The strategy's reliance on the ratio of the selectionrates (H/A) to indicate an employer's discriminatory practices isbased on the assumption that the ratio of the (Q/A) proportions is1.0. This is always an untested and questionable assumption. Themost accurate single measure of the discriminatory nature of theemployer's hiring process, the ratio of the proportion of qualifiedapplicants who are hired, (H/Q), is not directly observable. Thus,to accurately detect and evaluate the adverse impact of the em-ployer's selection process requires the use of both disaggregatedmeasures of the selection process, e.g., (H/S), and direct qualitativeanalyses of the procedures used in the process. 24

ANALYTICAL PROCEDURES TO DETECT DISCRIMINATION

The third issue in the use of quantitative information in employ-ment discrimination cases concerns how the information is prop-erly analyzed to indicate discrimination. When quantitative infor-mation indicates differential treatment by an employer of

23 The exceptions to the use of the "bottom line" strategy, see note 22 supra, would notmake the employer in Case 2 subject to evaluation of the components of his selection pro-cess. Thus the employer's discriminatory practice of hiring blacks who have passed the se-lection procedures at a lower rate than whites would go undetected.

24 To simplify the illustration, the hiring model described above includes only one term toindicate selection testing (S/Q). The model can easily be expanded to include separate termsfor each stage of the selection process, e.g., interviews, paper and pencil tests.

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individuals who differ only with respect to race, it is a reasonableassumption that discrimination has occurred. The basis of this as-sumption is that an individual's race has no effect on productivity.Therefore, the differential treatment by race cannot be motivatedby an acceptable economic reason. 25 It is so implausible that racecan determine a worker's productivity that race is not liable to be a"bona fide occupational qualification" (BFOQ).26 A similar categor-ical determination of irrelevance to productivity cannot be madefor the characteristics of sex and age. Both sex and age can be as-sumed to influence productivity in certain instances. As a result,the equal employment opportunity guidelines on sex and age dis-crimination include discussions indicating when these characteris-tics constitute a BFOQ.27 Since age and sex, but not race, can in-fluence productivity, differential treatment by sex and age mayhave an economically rational basis. This produces a difference inthe type of quantitative information which is appropriate to indi-cate sex and age discrimination from the type of evidence appro-priate to indicate race discrimination.

The basic distinction in types of evidence involves the analyticalconcept of an "intervening variable." An intervening variable is avariable which is the source of a relationship between two othervariables. As an illustration, consider a relationship between raceand productivity. In this hypothetical example, it is found thatblacks, on average, have lower productivity than whites. It is alsofound that the level of education of a worker is positively relatedto productivity and that the average level of education is lower forthe group of blacks. If the relationship between race and produc-tivity is examined separately for each level of education, it is foundthat there is no difference in productivity between blacks andwhites.2 The productivity difference between blacks and whites

25 Defining economic discrimination as existing when factors, other than productivity, de-

termine a worker's pay or the probabilities of hiring, promotion or dismissal is common inthe social scientific literature on discrimination. See Aigner & Cain, Statistical Theories ofDiscrimination in Labor Markets, 30 INDUS. & LABOR REL. Rev. 175 (1977).

26 The differential treatment of a minority group as a result of a "business necessity" doesnot contradict this. See Griggs v. Duke Power Co., 401 U.S. 424 (1971). The concept of abusiness necessity to hire a disproportionate number of whites results from an insufficientnumber of black applicants having the requisite qualifications for the particular job.

27 See Equal Employment Opportunity Commission, 29 C.F.R. § 1604.2 (1979), for theguidelines on sex as a BFOQ. See EEOC, Interpretations, 29 C.F.R. § 860.102 (1979), fordiscussion of age as a BFOQ relating to the Age Discrimination in Employment Act of 1967.

28 This technique of examining a relationship between two variables while holding con-stant the value of a third variable is called "statistical control." Discussions of statisticalcontrol can be found in most elementary statistical texts.

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has been explained by the racial difference in educational attain-ment. Education is the intervening variable in the relationship be-tween race and productivity.

For a characteristic such as sex or age to be a BFOQ, there mustbe a strong relationship between the characteristic and the jobqualification which is the intervening variable in the relationshipbetween the characteristic and productivity.29 The strength of therelationship between the characteristic and the intervening varia-ble can indicate the degree to which the use of the characteristic asa BFOQ is valid.

In certain instances, sex and age can be considered determinantsof productivity because there is a strong relationship between sexor age and a qualification for the job.30 In such instances it is nec-essary to evaluate whether differential treatment in wage, hiring,promotion or dismissal rates is truly a result of the relationshipbetween the characteristic and the job qualification, or if it is aresult of discrimination. Quantitative analyses can be useful inmaking this distinction.

As an example of this type of analysis, consider a hypothetical

2' As an example, consider the relationships between an actor's age, athletic ability andthe quality of a performance as Tarzan of the Jungle. The quality of the performance isnegatively related to an actor's age. This is the result of the reasonably strong relationshipbetween age and the intervening variable, athletic ability. As a result of the relationshipbetween the true occupational qualification of athletic ability and the characteristic of age,youth is considered a BFOQ to portray Tarzan. See EEOC, Interpretations, 29 C.F.R. §860.102e (1979).

30 If there is a "perfect" relationship between the characteristic and the job qualification,then the characteristic itself is a valid BFOQ. For example, the qualification for modelingwomen's fashions is a female anatomy, therefore, being female is a BFOQ. At the otherextreme there are job qualifications which have no relationship to sex, e.g., cognitive ability.Between these extreme examples is a variety of job qualifications which exhibit some rela-tionship to sex, e.g., physical strength. There is no standard ruling by the courts as to howstrong a relationship between a characteristic and a job qualification there must be beforethe characteristic is defined as a BFOQ. In Rosenfeld v. Southern Pac. Co., 444 F.2d 1219(9th Cir. 1971), that court held that "sexual characteristics, rather than characteristics thatmight. . . correlate with a particular sex, must be the basis of the BFOQ exception." Id. at1225. There the court indicated that only when there is a perfect relationship between thecharacteristic (sex) and the job qualification is a BFOQ acceptable. Compare this rulingwith the ruling in Massachusetts Bd. of Retirement v. Murgia, 427 U.S. 307 (1976). In thatcase the Supreme Court accepted the characteristic of age as a BFOQ in lieu of the actualjob qualification of physical fitness. The Court stated:

That the State chooses not to determine fitness more precisely through indi-vidualized testing after age 50 is not to say that the objective of assuring physi-cal fitness is not rationally furthered by a maximum age qualification.

Id. at 316. Thus in Murgia, a less than perfect relationship between a characteristic (age)and a job qualification (physical fitness) was accepted as a basis of making youth (being lessthan 50 years old) a BFOQ.

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employer who is hiring for a particular job in the company. Ananalysis of the job indicates that there are three qualifications nec-essary for success in the position. The employee must (1) be atleast a high school graduate, (2) be willing to stay on the job forthe next five years, and (3) be able to lift a 100-pound weight. Theemployer decides that each of these qualifications should beweighted equally in determining expected productivity, and there-fore the probability of hiring an applicant. The simple decision-making model used by the employer to determine who should behired can be described by the following formula:

HIRE = (GRAD/3) + (STAY/3) + (LIFT/3)

where:HIRE=the probability that the applicant is hired3"GRAD= 1, if the applicant has at least a high school education

0, if the applicant has less than a high schooleducation

STAY=the probability that the applicant is willing to stay onthe job for the next five years. This probability is cal-culated as [1-(applicant's age/70)]

LIFT=1, if the applicant is able to lift a 100-pound weight0, if the applicant is not able to lift a 100-poundweight.

The employer assumes that women, on the average, cannot passthe weightlifting test. The employer also knows that theprobability of a new employee staying on the job for the next fiveyears is negatively related to the employee's age, as described bythe formula: STAY= [1-(applicant's age/70)]. Given thisknowledge, the employer wants to determine whether being maleand young are BFOQs.

To resolve this question empirically, the employer analyzes thedata available from his pool of applicants for the job. A descriptionof the applicant pool is presented in Table II. Each applicant's sex,age, educational attainment, whether the applicant passed theweightlifting test, and the probability of staying on the job for fiveyears is reported in Table II. In addition, the probability of hiring

31 Viewing the decision of hiring a particular applicant as a probability, rather than adichotomous choice of hire or not hire, has certain heuristic value. However, the decision-making model described in the example may be altered to indicate whether or not an appli-cant is hired without affecting the conclusions derived from the example. Alternatively, theprobability of hiring (HIRE) can simply be thought of as representing the rank ordering ofapplicants for the position; with the greater the value of HIRE, the higher the rank of theapplicant.

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the applicant (HIRE), calculated using the decisionmaking modeldescribed above, is reported.

TABLE II

APPLICANT POOL

Job Qualifications

H.S. Passed Probability HiringGrad. Lift Test of Staying Probability

Applicant Sex Age (GRAD) (LIFT) 5 Years (HIRE)(STAY)

1 M 35 YES YES .50 .832 M 35 YES NO .50 .503 M 40 NO YES .43 .484 F 40 YES NO .43 .485 F 40 YES YES .43 .816 M 45 NO YES .36 .457 F 45 NO NO .36 .128 F 50 NO NO .29 .109 F 55 NO NO .21 .07

10 M 60 NO YES .14 .3811 F 60 NO NO .14 .0512 M 65 NO YES .07 .3613 F 65 YES NO .07 .3614 F 65 YES YES .07 .6915 M 70 YES NO .00 .3316 M 70 YES YES .00 .67

Using these data, the employer calculates the averageprobability of hiring a man or a woman. The average probability ofhiring a male applicant is 0.5; the average probability of hiring afemale applicant is 0.33. Since the probability of hiring is based onthe expected productivity in the job, this sex difference in the hir-ing probability represents an expected sex difference inproductivity.

To determine whether this sex difference is adequate groundsfor making being male a BFOQ, the employer needs to determinethe cause of the sex difference in the hiring probability (expectedproductivity). To do this, the employer analyzes the data for rela-tionships between sex and the three job qualifications (GRAD,STAY, and LIFT). An appropriate technique for this is correla-tional analysis.3 2 The correlation between sex and educational at-

32 A correlation coefficient indicates the strength of the relationship between two vari-

ables. The coefficient has a range of values from -1.0 to + 1.0. A value of -1.0 indicates aperfect negative relationship between two variables; a coefficient equal to + 1.0 indicates aperfect positive relationship. A correlation coefficient equal to 0.0 indicates the completeabsence of a relationship between two variables. A description of how correlation coefficientsare calculated can be found in most statistics texts.

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tainment (GRAD) and the correlation between sex and theprobability of staying on the job for five years (STAY) are both0.0, indicating there is no sex difference in either of these qualifica-tions.3 3 The correlation between sex and passing the weightliftingtest (LIFT) is 0.5, indicating a moderate sex difference in thisqualification."4 The ability to pass the weightlifting test is the in-tervening variable between sex and the hiring probability whichcauses the sex difference in that probability. This can be formallyshown by comparing the correlation coefficient between sex andthe hiring probability (HIRE) with the correlation between sex andthe hiring probability when differences in the ability to pass theweightlifting test are controlled for. This correlation between sexand the hiring probability after taking into account the weightlift-ing qualification is called a partial correlation. 5 The correlationbetween sex and the hiring probability is 0.34, indicating that fe-male applicants are less likely to be hired than male applicants.After controlling for whether or not an applicant passed the weigh-tlifting test, the partial correlation between sex and the hiringprobability is 0.0. This indicates that the sex difference in the hir-ing probability is completely due to the sex difference in the abilityto pass the weightlifting test.

The moderate strength of the relationship between sex and thejob qualification of passing the weightlifting test (the correlationcoefficient of 0:5) would make it difficult for the employer to sub-stantiate a claim that being male is a BFOQ for the job. Womenshould not be categorically excluded from the applicant pool.

A similar analysis is appropriate to determine whether youth is aBFOQ. Using the same data for the applicant pool, there is a corre-lation of -0.24 between the applicant's age and the probability ofbeing hired. This indicates that the older an applicant is, the lower

" The lack of relationships between sex, educational attainment and the probability ofstaying on the job for five years was built into the data to simplify the example. Sex differ-ences in these qualifications and relationships among the qualifications could exist in datafor an actual applicant pool. If such relationships do exist, the basic analytical techniqueused in the example is still appropriate, although the calculations become considerably morecomplex.

"' Another method of noting the sex difference in the weightlifting job qualification is tocalculate the percentages of male and female applicants who passed the test. Seventy-fivepercent of the male applicants passed the test compared to only twenty-five percent of thefemale applicants.

36 The value of a partial correlation coefficient has the same interpretation as the value ofa simple correlation coefficient, see note 32 supra, i.e., its range of values is from -1.0 to+1.0, with 0.0 indicating the lack of a relationship after controlling for the interveningvariable.

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is the probability of being hired. An analysis of the correlationsbetween age and the three job qualifications reveals correlation co-efficients of 0.0 between age and educational attainment (GRAD)and between age and whether the applicant passed the weightlift-ing test (LIFT). There is a correlation coefficient of -1.0 betweenage and the probability that the applicant will stay on the job forfive years.3

In this extreme example, youth completely determines this jobqualification. That the age difference in the probability of beinghired results from age determining this job qualification can bedemonstrated by the partial correlation between age and the hiringprobability controlling for the probability of staying on the job forfive years (STAY). This partial correlation equals 0.0, indicatingthat the simple correlation between age and the probability of be-ing hired (-0.24) is a result of the intervening variable STAY.

Therefore, youth is a BFOQ because it completely determinesone of the job qualifications. Given that staying on the job for fiveyears is truly a determinant of productivity and the probability ofstaying is accurately described by the formula used to calculateSTAY, the employer might properly use youth as a BFOQ. s7

The results from this type of analysis can be used as quantita-tive evidence of the validity or invalidity of the use of a particularcharacteristic as a BFOQ. An additional use of this analysis is toprovide direct quantitative evidence of discrimination in situationswhere an applicant's characteristic is related to a qualification ofthe job and is the object of an employer's discrimination.

36 As in the example of sex differences, the lack of a relationship between age and both

educational attainment and whether the applicant passed the weightlifting test is an artifi-cial construction used to simplify the example. The perfect relationship between age and theprobability of staying on the job for five years is also artificially built into the data. Itspurpose is to indicate an extreme example of when a characteristic of an applicant and a jobqualification are perfectly related. It is this type of perfect relationship that was considerednecessary for a characteristic to be a BFOQ in Rosenfeld v. Southern Pac. Co., 444 F.2d1219 (9th Cir. 1971).

37 With youth as a BFOQ, the employer's decisionmaking model may be described by theformula:

HIREt = (GRAD/3) + ((70 - AGE)/210) + (LIFT/3)where AGE is the applicant's age. In this example, HIREr, the probability an applicant ishired when youth is a BFOQ, equals HIRE as reported in Table II. This is because STAY iscompletely determined by the applicant's age. It is worth noting that using youth as aBFOQ does not categorically exclude an applicant from being hired because of the appli-cant's age. It merely allows the explicit use of the applicant's age in determining whetherthe individual should be hired. Categorical exclusion on the basis of a characteristic which isa BFOQ is only proper when that characteristic is the sole determinant of expectedproductivity.

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To illustrate this use of the analysis, it is necessary to elaborateupon the example of the employer who is hiring for a job. The em-ployer's decisionmaking model described above is objectivelynondiscriminating because neither sex nor age is explicitly in-cluded as a determinant of the probability of hiring. This decision-making model and the resulting probability that an applicant ishired are "sex-blind" and "age-blind."

As alternate decisionmaking models consider the following twoformulas used to calculate the probability of hiring an applicant:

HIRE* = (GRAD/3) + (STAY/3) + (LIFT/6) + (SEX/6)

and

HIRE** = (GRAD/4) + (STAY/4) + (LIFT/4) + (SEX/4)

where GRAD, STAY and LIFT are defined as above, and SEXequals 1 if the applicant is male and 0 if the applicant is female.The decisionmaking model used to calculate HIRE* weights theweightlifting qualification only half as much as the model deter-mining the probability HIRE. This fact alone would decrease thesex difference in the probability of being hired. However, the ap-plicant's sex is included as a qualification, weighted to be half asimportant as educational attainment and willingness to stay on thejob for five years. The probability HIRE** is determined by a deci-sionmaking model which places equal weight on the three job qual-ifications and the applicant's sex. Using these decisionmakingmodels, the resulting probability that an applicant is hired is di-rectly determined by the applicant's sex. Both models discriminateagainst female applicants by lowering their probability of beinghired regardless of their qualification for the job (expectedproductivity).

Since the decisionmaking model used by an employer is rarely asexplicitly stated as in the formula used to calculate HIRE, HIRE*or HIRE**, the proof of discrimination is the determination ofwhich model the employer uses in his hiring decisions. Whetherthe employer uses the nondiscriminatory model which producesthe nondiscriminatory hiring probability HIRE or either of the twodiscriminatory models can be determined by examining the partialcorrelation between sex and the hiring probability after controllingfor the job qualifications. A comparison of the partial correlationsresulting from each of the three decisionmaking models is reportedin Table III.

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TABLE III

RELATIONSHIP BETWEEN SEX AND HIRING PROBABILITYWITH THREE DECISIONMAKING MODELS

Nondiscriminatory Discriminatory ModelsModel

HIRE HIRE* HIRE**

Correlation BetweenSex and HiringProbability .34 .55 .74

Partial CorrelationBetween Sex and HiringProbability Controllingfor WeightliftingQualification .00 .38 .63

Percent of SexDifference inHiring ProbabilityDue to Employer'sDiscrimination 0% 69% 85%

As noted above, with the nondiscriminatory probability HIRE,the correlation between sex and the hiring probability is totally aresult of the sex difference in the weightlifting job qualification.Thus when the relationship between sex and the hiring probabilityis controlled for differences in this qualification, the partial corre-lation of sex and the hiring probability is zero.

The lack of a relationship between sex and the hiringprobability, after taking into account the sex difference in theweightlifting qualification, is not evident when either of the twodiscriminatory decisionmaking models is used by the employer.When the discriminatory hiring probability HIRE* is the appli-cant's chance of employment, a positive correlation between sexand the hiring probability remains after controlling for differencesin qualifications. When the decisionmaking model used by the em-ployer produces the hiring probability HIRE**, the evidence ofdiscrimination is even more pronounced. The partial correlationbetween sex and hiring probability after controlling for sex differ-ences in job qualifications is 0.63. Eighty-five percent (0.63/0.74) ofthe sex difference in the probability of being hired is due to factorsunrelated to job qualifications, i.e., it is a result of discrimination.

The litigant in an employment discrimination case who candemonstrate that a relationship exists between sex (age, race,ethnicity) and the hiring (wage, promotion, dismissal) rate, aftercontrolling for differences in job qualifications, has demonstratedthat the employer is using a discriminatory decisionmaking model.

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In practice, the analysis required for this demonstration can beconsiderably more complex than the example presented here. Pro-ductivity is determined by many qualifications, some of which maybe intervening variables between productivity and characteristicsof sex or age. A further complication is the general lack of mea-surement of the actual qualifications for a job. Few employers havemore than a vague notion of what determines a worker's productiv-ity in a particular job. However, as a result of requirements foremployment test validation in the U.G.E.S.P., objective job analy-ses to determine the qualifications will become more common.3 8

Once these qualifications are defined and applicants are properlymeasured to determine whether they possess the qualifications, anobjective determination of discrimination as illustrated in the anal-ysis may become a common evidential procedure.

CONCLUSION

The three issues discussed in this paper illustrate the potentialuse and misuse of quantitative information in employment dis-crimination cases. It should be evident from these illustrations thatlitigants in employment discrimination cases need to understandthe basic logic of statistical inference, the proper measures to indi-cate discrimination and the analytic procedures which can be usedto demonstrate discrimination.

There is no need for lawyers to become statisticians. The actualanalysis and presentation of quantitative information should beleft to experts. However, proper evaluation of quantitative infor-mation as probative to employment discrimination is an appropri-ate task for the legal community.

s See §§ 14B2 & 14D2 of the U.G.E.S.P., 43 Fed. Reg. 38,300, 38,307 (1978).

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